Question

Prove: A\(B∪C)=(A\B)∩(A\C)

Answer 1

Indirect proof means assume the opposite and find a contradiction.

If a + c > b + c then a > b

Answer 2

assume that a < b

Then there exists a positive number such that a+X = b

Now, plug this value into the first part...
a + c > (a+X) + c

So
(a + c) > (a + c) + X

So, a number PLUS a positive number is less than the number itself? This is a contradiction.

So it is not the case that a < b

Part 2: assume a = b
a + c > a + c

No number (a+c) can be greater than itself, so this is a contradiction.

So, it is not the case that a = b

So, a is not <= b

So a > b

By the way, it doesn't matter if a, b or c = 0, the simply can't equal each other.

Answer 3

why these answers get more and more random as the day goes on

Answer 4

\[ \large A\setminus(B\cup C)=A\cap(B\cup C)^c
=A\cap(B^c\cap C^c) \]
\[ \large A\cap A\cap B^c\cap C^c=(A\cap B^c)\cap(A\cap C^c)= \]